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Dynamics of a Class of Extended Duffing–Van Der Pol Oscillators: Melnikov’s Approach, Simulations, Control over Oscillations

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  • Nikolay Kyurkchiev

    (Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 24, Tzar Asen Str., 4000 Plovdiv, Bulgaria
    Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 8, 1113 Sofia, Bulgaria)

  • Tsvetelin Zaevski

    (Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 8, 1113 Sofia, Bulgaria
    Faculty of Mathematics and Informatics, Sofia University “St. Kliment Ohridski”, 5, James Bourchier Blvd., 1164 Sofia, Bulgaria)

  • Maria Vasileva

    (Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 24, Tzar Asen Str., 4000 Plovdiv, Bulgaria)

  • Vesselin Kyurkchiev

    (Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 24, Tzar Asen Str., 4000 Plovdiv, Bulgaria)

  • Anton Iliev

    (Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 24, Tzar Asen Str., 4000 Plovdiv, Bulgaria)

  • Asen Rahnev

    (Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 24, Tzar Asen Str., 4000 Plovdiv, Bulgaria)

Abstract

The Duffing–van der Pol oscillator is a very prominent and interesting standard model. There is a substantial body of varied literature on this topic. In this article, we propose a new class of oscillators by adding new factors to its dynamics. Investigations in light of Melnikov’s approach are considered. Several simulations are composed. A few specialized modules for testing the dynamics of the hypothetical oscillator under consideration are also given. This will be an essential component of a much broader Web-based scientific computing application that is planned. Possible control over oscillations: approximation with restrictions is also discussed; some probabilistic constructions are also presented.

Suggested Citation

  • Nikolay Kyurkchiev & Tsvetelin Zaevski & Maria Vasileva & Vesselin Kyurkchiev & Anton Iliev & Asen Rahnev, 2025. "Dynamics of a Class of Extended Duffing–Van Der Pol Oscillators: Melnikov’s Approach, Simulations, Control over Oscillations," Mathematics, MDPI, vol. 13(14), pages 1-21, July.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:14:p:2240-:d:1699112
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    References listed on IDEAS

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    1. Malik Zaka Ullah & Ramandeep Behl & Ioannis K. Argyros, 2020. "Some High-Order Iterative Methods for Nonlinear Models Originating from Real Life Problems," Mathematics, MDPI, vol. 8(8), pages 1-17, July.
    2. Nikolay Kyurkchiev & Tsvetelin Zaevski & Anton Iliev & Vesselin Kyurkchiev & Asen Rahnev, 2025. "Dynamics of a Class of Chemical Oscillators with Asymmetry Potential: Simulations and Control over Oscillations," Mathematics, MDPI, vol. 13(7), pages 1-21, March.
    3. Messias, Marcelo & Cândido, Murilo R., 2024. "Analytical results on the existence of periodic orbits and canard-type invariant torus in a simple dissipative oscillator," Chaos, Solitons & Fractals, Elsevier, vol. 182(C).
    4. Nikolay Kyurkchiev & Tsvetelin Zaevski & Anton Iliev & Todor Branzov & Vesselin Kyurkchiev & Asen Rahnev, 2024. "Dynamics of Some Perturbed Morse-Type Oscillators: Simulations and Applications," Mathematics, MDPI, vol. 12(21), pages 1-23, October.
    5. Taras Lukashiv & Yuliia Litvinchuk & Igor V. Malyk & Anna Golebiewska & Petr V. Nazarov, 2023. "Stabilization of Stochastic Dynamical Systems of a Random Structure with Markov Switches and Poisson Perturbations," Mathematics, MDPI, vol. 11(3), pages 1-22, January.
    6. Nikolay Kyurkchiev & Tsvetelin Zaevski & Anton Iliev & Vesselin Kyurkchiev & Asen Rahnev, 2025. "Investigations of Modified Classical Dynamical Models: Melnikov’s Approach, Simulations and Applications, and Probabilistic Control of Perturbations," Mathematics, MDPI, vol. 13(2), pages 1-21, January.
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